Uniform stability of nonlinear time-varying impulsive systems with eventually uniformly bounded impulse frequency

TL;DR

This paper establishes new stability conditions for nonlinear, time-varying impulsive systems, broadening the classes of impulse sequences for which stability can be guaranteed, and strengthening existing results in the literature.

Contribution

It introduces generalized stability criteria that encompass broader impulse frequency classes and unify various stability notions for impulsive systems.

Findings

Stability bounds are uniform with respect to initial time and impulse sequence classes.

Broader classes of impulse-time sequences are considered, beyond fixed or average dwell times.

Results strengthen and unify existing stability conditions for impulsive systems.

Abstract

We provide novel sufficient conditions for stability of nonlinear and time-varying impulsive systems. These conditions generalize, extend, and strengthen many existing results. Different types of input-to-state stability (ISS), as well as zero-input global uniform asymptotic stability (0-GUAS), are covered by employing a two-measure framework and considering stability of both weak (decay depends only on elapsed time) and strong (decay depends on elapsed time and the number of impulses) flavors. By contrast to many existing results, the stability state bounds imposed are uniform with respect to initial time and also with respect to classes of impulse-time sequences where the impulse frequency is eventually uniformly bounded. We show that the considered classes of impulse-time sequences are substantially broader than other previously considered classes, such as those having fixed or (reverse) average dwell times, or impulse frequency achieving uniform convergence to a limit (superior or inferior). Moreover, our sufficient conditions are not more restrictive than existing ones when particularized to some of the cases covered in the literature, and hence in these cases our results allow to strengthen the existing conclusions.